Beijing Institute of Mathematical Sciences and Applications

In one sentence, the teaching of deformation theory is that, in characteristic zero, every equation describing deformation of some object can be rewritten as an equation $(d+a)^2=0$ with an appropriate understanding of what precisely $d$ and $a$ are. These are Maurer-Cartan equations in differential graded (or dg) Lie algebras. The other, seeminly unrelated reason to study dg Lie algebras is rational homotopy theory --- by works of Quillen and Sullivan, the homotopy category of (nilpotent and connected) dg Lie algebras is the so-called rational homotopy category, which is, roughly, category that knows everything about homotopy theory of topological spaces "modulo torsion". In this course we will study the homotopy category of dg Lie algebras with the goal to application to these two subjects --- deformations and rational homotopies, and with the emphasis on examples. At the culmination point of a course we are going to apply the techniques studied to an algebraic problem that is seemingly unrelated to homotopy theory at all --- we will prove a theorem that a nilpotent Lie algebra could be reconstructed from knowing its universal enveloping as just an associative algebra.